Problem 102 - A Graphical Approach to Precalculus with Limits

Question

Geometry Determine graphically whether it is possible to construct a cylindrical container, including the top and bottom, with volume 38 cubic inches and surface area 38 square inches.

Step-by-Step Solution

Answer

No, the graph indicates no intersection, so such a container is not possible.

Step 1: Understand the Problem

We are tasked with verifying if it's possible to construct a cylindrical container with both a given volume and surface area of 38 cubic inches each. To solve this graphically, we need equations for both values in terms of a common variable.

Step 2: Recall Formulas

For a cylinder, the volume \( V \) is given by \( V = \pi r^2 h \), and the surface area \( A \) is \( A = 2\pi r(h + r) \), where \( r \) is the radius and \( h \) is the height.

Step 3: Set Up Equations

Given the volume equation \( \pi r^2 h = 38 \) and the surface area equation \( 2\pi r(h + r) = 38 \), isolate \( h \) in terms of \( r \) from the volume equation: \( h = \frac{38}{\pi r^2} \).

Step 4: Substitute and Rearrange Surface Area Equation

Substitute \( h = \frac{38}{\pi r^2} \) into the surface area equation to get \( 2\pi r\left(\frac{38}{\pi r^2} + r\right) = 38 \). Simplify and rearrange to create a new equation in terms of \( r \).

Step 5: Graph Both Equations

Plot the equations derived from the volume and the surface area on a graph with the radius \( r \) on the x-axis. This will visualize whether there is a common \( r \) that satisfies both equations.

Step 6: Analyze Graph Intersection

Check the graph to determine whether the two plotted curves intersect at any point. An intersection means there is a common \( (r, h) \) pair that satisfies both equations.

Key Concepts

Volume of a Cylinder

The formula for the volume of a cylinder is a fundamental concept in geometry. Understanding how to calculate it is essential when dealing with cylindrical shapes. The volume is the space contained within the cylinder and is found using the formula:\[ V = \pi r^2 h \]where:

• \( V \) is the volume of the cylinder,
• \( r \) is the radius of the base, and
• \( h \) is the height of the cylinder.

To find the volume, you multiply the area of the base, which is a circle, by the height of the cylinder. In practical problems, you often need to solve for one of the variables, such as finding the height when given the volume and radius. This requires rearranging the formula:\[ h = \frac{V}{\pi r^2} \]When working with these equations, remember to keep your units consistent. Whether you're working with inches, centimeters, or another unit, the final volume will be in cubic units since it accounts for space in three dimensions.

Surface Area of a Cylinder

The surface area of a cylinder involves the total area that the surface of the object occupies. This includes the areas of both circular ends and the side of the cylinder, known as the lateral surface. The formula to calculate this is:\[ A = 2\pi r(h + r) \]where:

• \( A \) is the surface area,
• \( r \) is the radius, and
• \( h \) is the height.

Breaking it down, the surface area combines the area of the two circles (top and bottom) and the rectangle that wraps around the side of the cylinder (lateral surface). Each circle has an area of \( \pi r^2 \), and the lateral surface, which can be "unwrapped" into a rectangle, has dimensions \( 2\pi r \times h \).The total formula hence includes:

• \( 2\pi r^2 \) for both circular bases, and
• \( 2\pi rh \) for the lateral surface.

Understanding this formula is crucial, especially when designing objects like cans, pipes, or any container that requires a protective coating or paint to cover its entire surface.

Graphical Solutions in Mathematics

Graphical solutions in mathematics provide a visual approach to solving equations and understanding relationships between variables. When you plot equations graphically, you can easily see where solutions exist, particularly intersection points.For the problem of determining if a cylindrical container can exist with both specified volume and surface area, you first need to express the volume and surface area formulas in terms of the radius \( r \).Next, by setting up and plotting these equations on a graph with \( r \) on the x-axis, you can visualize their behaviors and check for intersections:

• If the graphs intersect, there is a potential solution where the radius and height satisfy both conditions.
• If they do not intersect, such a cylinder is not possible with the given constraints.

This method helps students grasp the concept of multiple variables and how they interact. Graphical solutions often provide an intuitive understanding that algebraic manipulation alone might not. They are particularly useful in complex problems where visualizing changes can make the solutions easier to interpret.